超空间中的广义Cauchy-Kovalevskaya扩张与平面波分解
Generalized Cauchy-Kovalevskaya extension and plane wave decompositions in superspace
摘要：
本文的目的是得到双轴Dirac算子在超空间中的一个推广的CK-扩张定理。在经典的交换情形下，这个结果可以表示为某些微分算子作用在单个初始函数上的Bessel型幂级数。在超空间的背景下，新的结构出现在负的甚至超维的情况下。在这些情况下，CK-扩张依赖于两个初始函数，两个微分算子的幂级数作用在这两个初始函数上。这些系列不仅是贝塞尔型，但它们产生了一个额外的结构方面的Appell多项式。这种模式也存在于Pizzetti公式的结构中，该公式用微分算子描述了超球上的积分。我们通过研究超球面上的广义CK-扩张分解成平面波来明确这个关系。此外，这些结果被应用于超空间中的柯西核到单演平面波的分解，这将有助于反演超Radon变换。
英文摘要：
The aim of this paper is to obtain a generalized CK-extension theorem in superspace for the bi-axial Dirac operator. In the classical commuting case, this result can be written as a power series of Bessel type of certain differential operators acting on a single initial function. In the superspace setting, novel structures appear in the cases of negative even superdimensions. In these cases, the CK-extension depends on two initial functions on which two power series of differential operators act. These series are not only of Bessel type but they give rise to an additional structure in terms of Appell polynomials. This pattern also is present in the structure of the Pizzetti formula, which describes integration over the supersphere in terms of differential operators. We make this relation explicit by studying the decomposition of the generalized CK-extension into plane waves integrated over the supersphere. Moreover, these results are applied to obtain a decomposition of the Cauchy kernel in superspace into monogenic plane waves, which shall be useful for inverting the super Radon transform.
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